Stochastic Thermodynamics of Social Imitation beyond Energetics

TL;DR

This work extends stochastic thermodynamics to social systems lacking energetic degrees of freedom by introducing a microscopically reversible two-state imitation model with group interactions (herding and anticonformity). It derives trajectory-level and ensemble fluctuation relations, including a detailed and integral fluctuation theorem for social entropy production $S_{ m tot}$ and a strong current fluctuation theorem that enables parameter inference from observed opinion currents. The authors uncover a rich nonequilibrium structure with second- and first-order phase transitions and demonstrate TUR and KUR trade-offs between dissipation, current precision, and dynamical activity, plus a symmetry-breaking framework for driven transitions between polarized and consensus states. The framework yields practical inference tools for estimating model biases from data and lays groundwork for applying energy-free stochastic thermodynamics to broader sociophysical contexts and networked systems.

Abstract

The development of stochastic thermodynamics during the last decades prompted the discovery of novel nonequilibrium relations refining our understanding of the second law in small fluctuating systems and its connection with information theory. A fundamental open question is whether these powerful tools can illuminate other areas of complex systems, such as social phenomena, where energy plays no fundamental role. Here we develop a framework that derives a ``second law" for social systems. Similarly to Landauer's principle, it constrains spontaneous changes in agent attributes (opinions, cultural traits, etc.) and their informational entropy. We apply this framework to toy agent-based models of social imitation with non-trivial phase diagrams. We demonstrate how cornerstone results -- fluctuation theorems, kinetic and thermodynamic uncertainty relations, and second-law-like inequalities -- emerge naturally in this context, even across symmetry-breaking transitions. These results reveal fundamental trade-offs in opinion currents arising from the competition between herding and anti-conformity. Moreover, they provide inference tools to extract model parameters from observations of stochastic changes in agents.

Figures (9)

• Figure 1: Illustration of herding and anticonformity mechanisms in the two reactions \ref{['eq:Model_Reaction_1']} (top) and \ref{['eq:Model_Reaction_2']} (bottom). The herding mechanism occurs when the $q$ agents are of opposite opinion, leading the selected agent to conform and change its opinion. Conversely, the anticonformity mechanism occurs when the $q$ agents share the same opinion as the selected agent, prompting it to differentiate and change its opinion. The example shown corresponds to the case of sampling without repetition with $q = 6$: a chosen agent (lighter color) is influenced by 6 distinct neighbor agents (darker color) to switch its initial state. Note that the interpretation of the parameter $q$ depends on the sampling scheme: without repetition, $q$ represents the exact number of distinct neighbors sampled; with repetition, $q$ denotes the total number of interactions.
• Figure 2: Stationary probability distribution coming from Eq. \ref{['eq:Stationary Distribution Analytical Solution']} taking $g(n)=\left[{n}/{(n-1)}\right]^q$ for $25$ different values of $\lambda$ equally distributed in the interval $\lambda\in[2.5,5.0]$. We observe how the system transits from a unimodal distribution centered at $N/2$ ($\lambda < 3$) to a bimodal distribution ($\lambda > 3$) with the peaks shifting towards the extremes $n \in \{0, N\}$ as $\lambda$ increases. The transition occurs at $\lambda_{\mathrm{c}} = 3$ as given by \ref{['eq:critical point asymmetric']}. Parameters: $N = 100, q = 2, \chi = \theta = 1$.
• Figure 3: Phase diagram of the model defined in Eqs. \ref{['eq:transition rates']}, in the $(\lambda, \chi)$ parameter space for $g(n) = [n/(N-1)]^q$, with $q=2$ and symmetric reactions ($\theta=1$). Regions where opinion $A$ is predominant ($\langle n \rangle_{\mathrm{st}} > N/2$) are shown in red, while those favoring opinion $B$ ($\langle n \rangle_{\mathrm{st}} < N/2$) are shown in blue. Each region is further subdivided into unimodal and bimodal zones, delimited by the critical curves $\chi_\pm(\lambda)$ (dashed lines), which converge to the critical point (black dot) at $(\lambda_{\mathrm{c}}, \chi_{\mathrm{c}}) = (3, 1)$ as predicted by Eq. \ref{['eq:critical point asymmetric']}. The line $\chi = 1$ corresponds to the symmetric case illustrated in Fig. \ref{['fig:stationary probability distribution']}, which contains the unimodal $\chi_{\rm u}$ (dash-dotted) and bimodal $\chi_{\rm b}(\lambda)$ (solid) transition lines. The insets show the corresponding Fokker--Planck potential $v(x)$ (see Appendix \ref{['sec:Stationary State apdx']}) at selected points in the phase diagram, illustrating how the absolute minimum (marked by a gray circle) determines the most probable stationary state.
• Figure 4: Illustration of a forward trajectory, $\gamma_{[0,\tau]}$, with $3$ jumps up ($k_i = +$ for $i=1,2, 4$) and $1$ jump down ($k_3 = -$), together with a driving protocol $\Lambda$ (top plot); and with the inverted protocol of the backward process, $\tilde{\Lambda}$ (middle plot). The corresponding time-reversed trajectory, $\tilde{\gamma}_{[0,\tau]}$, with the inverse jumps ($\tilde{k}_i = -$ for $i=1,2,4$ and $\tilde{k}_3 = +$) in the backward process $\tilde{\Lambda}$ is illustrated in the bottom plot. For obtaining Eq. \ref{['eq:DFT']} we compare the probabilities of trajectories in top and bottom plots.
• Figure 5: Ensemble thermodynamic analysis for $q = 2$ and $\theta = 1$. Panels (a--c) show intensive stationary quantities as functions of $\lambda$ with $\chi = 1$: (a) probability current $\langle \dot{I}_1\rangle_{\mathrm{st}}/N$, (b) entropy production rate $\langle \dot{S}_{\mathrm{tot}}\rangle_{\mathrm{st}}/N$ from Eq. (\ref{['eq:Stotst']}), and (c) dynamical activity $\langle K\rangle_{\mathrm{st}}/N$. These curves are calculated with the stationary distribution Eq. (\ref{['eq:Stationary Distribution Analytical Solution']}). Exact results for system sizes $N\in\{10^2,10^3,10^4\}$ are shown alongside the mean-field solution (black dash-dotted line, see App. \ref{['sec:Mean-field analytical results']}). Vertical dotted and dashed lines mark the equilibrium point ($\lambda_{\mathrm{eq}}=1$) and critical point ($\lambda_{\mathrm{c}}=3$), respectively. Inset in panel (a): dependence of $\langle \dot{I}_1\rangle_{\mathrm{st}}/N$ on $\chi$ at fixed $\lambda=5$, indicating $\chi_{-}$ (blue) and $\chi_{+}$ (red).